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Abstract

In plasma physics, it is often necessary to study the motion of a charged particle
in a magnetic field, modelling the effect of collisions by additive random forces.
Following Langevin, the force is taken to be the sum of a friction term, proportional to
the velocity, and a fluctuating component referred to as “noise”. This stochastic term is
assumed to be a Gaussian stationary process with vanishing mean and a covariance
matrix characterized by correlation times which may vanish (“white noise”) or be finite
(e.g. “coloured noise”). This problem, for the case of a uniform field, leads to two
systems of equations, one for the ID longitudinal motion along the direction of the field
and another for the 2D transverse motion perpendicular to it. These two systems can be
studied independently. Because of the linearity of the equations, formal solutions are
written and expressions for the expectation values of powers of the position and/or the
velocity of the particle are obtained. For white noise, the problem has been studied
analytically and numerically. However, the case of a stochastic term with finite
correlation time has not been considered in detail.
Here, the influence of non-white noises is studied and numerical simulations
based on two different approaches are presented. In the first approach, by adding an equation for the evolution of the stochastic term, the problem is reduced essentially to
that of white noise. However, this technique is not to be adopted in general since it can
only reproduce specific kinds of noise. In the second approach the forcing term is
modelled by a Fourier series with random, uniformly distributed, phases and may be
considered as more general. Simulations with the latter method are presented for a
random force with the correlation function of coloured, Gaussian or Lorentzian noise. To
judge the extracted accuracy from the computational approaches, the numerical results
have also been compared with available analytical solutions. In addition, an extensive
parametric study has been performed with respect to various involved parameters
including the correlation time and the friction coefficient.
This numerical investigation serves for validating and benchmarking purposes of
the implemented computational schemes and will be used in more complex problems.
The problem of motion of a charged particle inside a space dependent electric field has
also been investigated. The governing system of equations is non-linear and can only be
solved numerically using the Fourier series approach because the explicit form of the
field is not available. Results show good qualitative agreement with benchmark problems.
In the future, random motion in electromagnetic fields and inhomogeneous
magnetic fields will also be considered.